I will follow my previous post.
My path to hunting and shooting in terms of inventory was gradual.
First firearms bought for my shooting was:
22lr pistol, 22lr rifle, 9x19 pistol.
For start, all was for purpose to learn shooting and training.
next (very soon) step was:
308 win, rifle, varmint type, and upland OU shotgun 12 GA
This was after i was confident I can hit the target with 22lr and was ready for hunting.
Rifle was used for training, and all round hunting, shotgun was used for small game and clay targets.
This covered a wide range of activities for some time. Big and small game hunting, birds, target shooting, clay pigeons, occasional competition, and training. (lets say, "quality trigger time")
next step was:
DA revolver, 357 mag, of course
and then there was a next step, then the next step, then the next step... etc so the inventory evolved, and was more specialized.
Coming back to original post, from OP, short and economical, one rifle approach to start hunting would be 308 win rifle, in my view.